If the government failed to make an interest payment, it would be very bad. That would cause government bonds to be downgraded, and be worth less money. Banks hold these bonds as capital, and bank rules say that you have to have a certain amount of capital. So if the government missed a payment, banks all over the world would see the value of their assets drop. They would have to store more capital, which would mean less lending, a lowering of the money multiplier and money supply, a credit crunch like the one in 2008, and another recession.
If the deadline comes without a deal, the government would probably keep the interest payments going and stop paying things like salaries of government workers. The damage from an US government default and downgrade would be immense. That is why bond prices have not changed much; the market knows that the administration knows this, and will not allow a default to happen.